Eshelby formalism for multi-shell nano-inhomogeneities

Nano-particles consisting of a core surrounded by multiple outer shells (multishell particles) are used as novel functional materials as well as stiffeners/toughners in conventional composites and nanocomposites. In these heterogeneous particles, the mismatch of thermal expansion coefficients and lattice constants between neighboring shells induces stress/strain fields in the core and shells, which in turn affect the physical/mechanical properties of the particles themselves and/or of the composites containing them. In this paper, we solve the elastostatic inhomogeneous inclusion problem of an infinite medium containing a multi-shell spherical particle when the eigenstrains are prescribed in the particle and in the multi-shells, and the inhomogeneity problem when an arbitrary remote stress field is applied to the infinite medium. The corresponding Eshelby and stress concentration tensors of the two problems are obtained and specialised to inhomogeneous inclusions in finite spherical domains with fixed displacement or traction-free boundary conditions. Finally, the Eshelby tensor of a spherical inhomogeneity with non-uniform eigenstrain is obtained and applied to quantum dots of uniform and non-uniform compositions.